Deep-sea drilling projects helped support the concept of plate tectonics in one major way. The drills brought back rocks from the bottom of the ocean that we had no way of accessing before. Scientists used radioactive dating to figure out how long ago the rocks formed. They eventually realized that the further you go away from the mid-ocean ridge (a long chain of underwater mountains on the seafloor), the older the rocks get. See this image for an illustration of this pattern:

This helped them realize that the mid-ocean ridges were places where new rock was forming, and was itself a plate boundary. At the mid-ocean ridge, molten magma rises to the ocean floor, cools, and begins to drift away from the ridge. Here is another image that illustrates this concept quite nicely:

This finding helped scientists understand that mid-ocean ridges are major "rock factories" where new crust is being formed on the bottom of the oceans.